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Mirrors > Home > MPE Home > Th. List > arwhoma | Structured version Visualization version GIF version |
Description: An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
arwrcl.a | ⊢ 𝐴 = (Arrow‘𝐶) |
arwhoma.h | ⊢ 𝐻 = (Homa‘𝐶) |
Ref | Expression |
---|---|
arwhoma | ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | arwrcl.a | . . . . . . 7 ⊢ 𝐴 = (Arrow‘𝐶) | |
2 | arwhoma.h | . . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | 1, 2 | arwval 17295 | . . . . . 6 ⊢ 𝐴 = ∪ ran 𝐻 |
4 | 3 | eleq2i 2881 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 ↔ 𝐹 ∈ ∪ ran 𝐻) |
5 | 4 | biimpi 219 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ∪ ran 𝐻) |
6 | eqid 2798 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
7 | 1 | arwrcl 17296 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat) |
8 | 2, 6, 7 | homaf 17282 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) |
9 | ffn 6487 | . . . . 5 ⊢ (𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V) → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶))) | |
10 | fnunirn 6990 | . . . . 5 ⊢ (𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)) → (𝐹 ∈ ∪ ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧))) | |
11 | 8, 9, 10 | 3syl 18 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∈ ∪ ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧))) |
12 | 5, 11 | mpbid 235 | . . 3 ⊢ (𝐹 ∈ 𝐴 → ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧)) |
13 | fveq2 6645 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐻‘𝑧) = (𝐻‘〈𝑥, 𝑦〉)) | |
14 | df-ov 7138 | . . . . . 6 ⊢ (𝑥𝐻𝑦) = (𝐻‘〈𝑥, 𝑦〉) | |
15 | 13, 14 | eqtr4di 2851 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐻‘𝑧) = (𝑥𝐻𝑦)) |
16 | 15 | eleq2d 2875 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹 ∈ (𝐻‘𝑧) ↔ 𝐹 ∈ (𝑥𝐻𝑦))) |
17 | 16 | rexxp 5677 | . . 3 ⊢ (∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧) ↔ ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦)) |
18 | 12, 17 | sylib 221 | . 2 ⊢ (𝐹 ∈ 𝐴 → ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦)) |
19 | id 22 | . . . . 5 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ (𝑥𝐻𝑦)) | |
20 | 2 | homadm 17292 | . . . . . 6 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → (doma‘𝐹) = 𝑥) |
21 | 2 | homacd 17293 | . . . . . 6 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → (coda‘𝐹) = 𝑦) |
22 | 20, 21 | oveq12d 7153 | . . . . 5 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → ((doma‘𝐹)𝐻(coda‘𝐹)) = (𝑥𝐻𝑦)) |
23 | 19, 22 | eleqtrrd 2893 | . . . 4 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
24 | 23 | rexlimivw 3241 | . . 3 ⊢ (∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
25 | 24 | rexlimivw 3241 | . 2 ⊢ (∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
26 | 18, 25 | syl 17 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 𝒫 cpw 4497 〈cop 4531 ∪ cuni 4800 × cxp 5517 ran crn 5520 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 domacdoma 17272 codaccoda 17273 Arrowcarw 17274 Homachoma 17275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-1st 7671 df-2nd 7672 df-doma 17276 df-coda 17277 df-homa 17278 df-arw 17279 |
This theorem is referenced by: arwdm 17299 arwcd 17300 arwhom 17303 arwdmcd 17304 coapm 17323 |
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